L2 approaches to holomorphic foliations

Abstract

Results on the L2 ∂̄ -cohomology groups are applied to holomorphic foliations with an emphasis on the cases with Levi flat hypersurfaces as stable sets. Nonexistence theorems are discussed for holomorphic foliations of codimension one on compact Kähler manifolds under some assumptions on geometric properties of the complement of stable sets. For the special cases such as ℂℙn, complex tori and Hopf surfaces, nonexistence, reduction and classification theorems will be proved. Closely related materials have been already discussed in Sect. 2.4., e.g. Theorem 2.79.